Gromov-Witten invariants of general symplectic manifolds

TL;DR

This paper introduces a new approach to defining Gromov-Witten invariants for any closed symplectic manifold, avoiding genericity assumptions and incorporating singular curves through a Banach orbifold framework.

Contribution

It develops a method to construct Gromov-Witten invariants using Banach orbifolds and Fredholm sections, applicable to all closed symplectic manifolds without genericity constraints.

Findings

Constructs Gromov-Witten invariants via localized Euler classes.

Avoids genericity assumptions in the invariants' definition.

Incorporates singular curves into the invariant formulation.

Abstract

We present an approach to Gromov-Witten invariants that works on arbitrary (closed) symplectic manifolds. We avoid genericity arguments and take into account singular curves in the very formulation. The method is by first endowing mapping spaces from (prestable) algebraic curves into the symplectic manifold with the structure of a Banach orbifold and then exhibiting the space of stable $J$-curves (``stable maps'') as zero set of a Fredholm section of a Banach orbibundle over this space. The invariants are constructed by pairing with a homology class on the locally compact topological space of stable $J$-curves that is generated as localized Euler class of the section.